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√2 is irrational prove

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√2 is irrational prove

Posted by Masoom Siddique 5 years, 11 months ago

CBSE > Class 10 > Mathematics

2 answers

Satyam Kumar 5 years, 11 months ago

Let us assume that √2 is rational no. √2=p/q. (q is not equal to 0) √2q=p. (S.B.S) 2q^2=p^2 Since 2 divides p square ,So 2 also divides p Also. p=2m. (M is some integer) p^2=4m^2. {Sbs} 2q^2=4m^2 q^2=2m^2 Since 2 divides q square so 2 also divides q -+But we know that rational numbers have only one common factor that is one So this contradicts our fact that under root 2 is rational So our exemption may be wrong, Hence √2 is rational number

Yogita Ingle 5 years, 11 months ago

Given √2 is irrational number.
Let √2 = a / b wher a,b are integers b ≠ 0
we also suppose that a / b is written in the simplest form
Now √2 = a / b ⇒ 2 = a² / b² ⇒ 2b² = a²
∴ 2b² is divisible by 2
⇒ a² is divisible by 2
⇒ a is divisible by 2
∴ let a = 2c
a² = 4c² ⇒ 2b² = 4c² ⇒ b² = 2c²
∴ 2c² is divisible by 2
∴ b² is divisible by 2
∴ b is divisible by 2
∴a are b are divisible by 2 .
this contradicts our supposition that a/b is written in the simplest form
Hence our supposition is wrong
∴ √2 is irrational number.

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